Answer: Grafting is a process by which two different, live plants are united to form one plant. For example, in fruit trees you can take a rootstock, consisiting of a young tree's roots and a short piece of trunk, and graft onto that a scion, which will become the top part of the tree--the part that produces edible fruit. That way, you get the benefit of choosing the best qualities for both the root system and the fruit.
As far as plants that you can graft indoors--you can experiment with different plants I suppose, but I don't know of any that are commonly done, except perhaps cactus. (If you've seen those funny little cactus for sale, that look like two different plants stuck together...well, that's what they are--two cacti grafted together.)
Grafting is generally done on woody plants. The key is to place the lateral meristem regions of both parts in contact. The lateral meristem is the actively growing part, located just beneath the bark. By carefully lining up the meristem, you also line up the phloem (food-conducting tissue) and xylem (water-conducting tissue).
Green, "herbaceous", non-woody plants don't have lateral meristems, they only have apical meristems (located at the tips of shoots.) I'm not aware of any way to graft two pieces of herbaceous plants. But that doesn't mean that a way doesn't exist! They have axillary buds located at the stem's nodes, so there may be a way to graft a new bud there.
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